There’s a little story about Hilbert that I told when I talked the other week; I can’t resist telling it again. In the 19 th century, a great German mathematician, George Cantor, really cleaned up this whole area of thinking about mathematics in a very remarkable way. If you come along with an infinite number of friends, you can all be accommodated. So this is the contradiction: you cannot systematically count all the never ending decimals. Cantor himself left mathematics for quite long periods as a result of depression and oppression through influential people who just didn’t like this type of thinking; they regarded it as a subversive part of mathematics. "He also knew that all nontrivial zeroes are symmetric with respect to the critical line x = 1/2. It may seem esoteric, but the understanding of infinity—and set theory—is vital … The first is the infinity of the mathematicians. And the third is the more religious idea of a transcendental infinity, a sort of “make me one with everything” type of idea. In some sense, not that much further – 10 to the 10 to the 50 metres, and if you want the first copy of the entire region inside our visible horizon today, you would have to go 10 to the 10 to the 119 metres. January 25, 2019. * We look at the different conceptions of 'infinity' that exist in mathematics, science and philosophy. There’s not a logical barrier or religious barrier to conceiving of things which are infinitely large. In one of Escher’s famous engravings, so-called cubic space division in English, you could imagine what’s going on is that you have an infinite network of crossing girders, which are going on for ever, in every direction, and if you walk around in that and then look around you, wherever you’re standing, you will see exactly the same situation, so there is no centre in this infinite network. Why this is particularly interesting is that suppose, given that you know there are all these unusual twin paradoxes of relativity in which you can send a twin off on a space flight and come back and find that the twin has aged quite differently from the identical twin that stayed at home, it’s not inconceivable that there might be some way in which you could send your laptop computer on a trip through space time so that when it returned, it had carried out an infinite number of computations, even though a similar computer back on Earth had just carried out a finite number. is probably the best way to get across the true meaning of the word "infinity". But if we go back further in time, we find this type of never-ending symbol appearing with religious significance – the famous cross of St Boniface, that you still sometimes see used as a religious symbol. He has also delivered lectures in a perhaps unique combination of locations including 10 Downing Street, Windsor Castle, the Vatican Palace and the Venice Film festival. This is something that’s been appreciated for a long time – I’ll give you several examples. However, in reality, the universe is not exactly the same in every place. The word is chucked about a lot by some people trying to express how huge something is ~ and although in physics and math it is understood what it means ~ most people just think ot it as meaning SO bloody many that it would take way too long to count. This is much closer to home in a way, easier to understand. There were reasons for this which we appreciate rather simply. But let's put it another way. So what happened in particle physics was that the quest to avoid infinities became a guiding principle in your search for the best or even the ultimate theory of elementary particles. Beyond the infinity known as ℵ 0 (the cardinality of the natural numbers) there is ℵ 1 (which is larger) … ℵ 2 (which is larger still) … and, in fact, an infinite variety of different infinities. How long will it take to cook a 12 pound turkey? You could keep on going in your spaceship and you would never reach the edge of such a universe, because it has no edge. In some places there are galaxies, in some there are not, so if you were to measure the expansion of the universe from one place to the other with fantastic precision, you would find it’s very slightly different. Next to it is a mathematical figure that looks the same. It’s one of these ones that expands forever, but it has a sort of kink in it and has started to accelerate again in the recent past. It is only a symbol that denotes an infinitely small (infinitesimal) or an infinitely large quantity. They felt that it had the potential to introduce a fallacy into mathematics, and the whole logical structure would then collapse. What this shows you I think is really that the question, whether the light is on or off at the end of the minute, is not a meaningful question. That’s it. Although you might not have guessed it, all fractions, like one number divided by another, 1 over 2, 3 over 4, 7 over 4, what mathematicians called the rational numbers, these are countable, so they’re no bigger than the ordinary numbers. Is evaporated milk the same thing as condensed milk? The second is a physical infinity, this question about whether you could do something in the world which would produce an infinite answer, result in an infinity being measured – very alarming and controversial. Wait, what Surgeons can Learn from Polar Explorers and Fighter Pilots old it. Build an entire infinite universe the true meaning of the founders of our picture of the largest numbers that could! 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